Additive Runge–Kutta methods for the resolution of linear parabolic problems

We study the consistency for general additive Runge–Kutta methods in the integration of linear nonhomogeneous problems, obtaining necessary and sufficient conditions of order p, for arbitrary values of p. We use this result joined to some A-stability conditions for developing a third order additive Runge–Kutta method of type fractional steps and we show its efficiency in the numerical integration of a two-dimensional evolutionary convection–diffusion problem.